1 Outline Part I: Theory steady state enzyme kinetics: a new - - PDF document

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Steady-State Enzyme Kinetics 1

A New 'Microscopic' Look at Steady-state Enzyme Kinetics

Petr Kuzmič BioKin Ltd. http://www.biokin.com SEMINAR: University of Massachusetts Medical School

Worcester, MA

April 6, 2015

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Steady-State Enzyme Kinetics 2

Outline

Part I: Theory steady state enzyme kinetics: a new approach Part II: Experiment inosine-5’-monophosphate dehydrogenase

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Steady-State Enzyme Kinetics 3

Enzyme kinetic modeling and its importance

WHAT CAN ENZYME KINETICS DO FOR US?

macroscopic

laboratory measurement

microscopic

molecular mechanisms

mathematical model

EXAMPLE: Michaelis-Menten (1913)

[substrate] initial rate maximum! rectangular hyperbola v = Vm S/(S+Km) MECHANISM:

substrate and enzyme form a reactive complex, which decomposes into products and regenerates the enzyme catalyst

The main purpose of enzyme kinetics is to elucidate microscopic molecular mechanisms of enzyme action and inhibition. It is a remarkable feat that we can go from observing events

• n a macroscopic scale (i.e. plate-reader well) to making

inference about events that evolve on a molecular scale. Another use of enzyme kinetics, no less important, is to measure quantitatively the strength enzyme binding with various ligands (substrates, inhibitors), as well as the reaction rates.

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Steady-State Enzyme Kinetics 4

Two types of enzyme kinetic experiments

• 1. “reaction progress” method:

time UV/Vis absorbance, A

• 2. “initial rate” method:

[Subtrate] rate = dA/dt @ t = 0 slope = “rate”

1. “reaction progress” method:

• mix enzyme and reactants
• monitor some experimental signal over time
• build mathematical models of the reaction time course
• see which of the models fits best
• 2. “initial rate” method:
• mix enzyme and reactants
• monitor some experimental signal over time
• compute the slope (reaction rate) at time = 0
• repeat at various concentrations of reactants
• build mathematical models of the initial rates changing with initial concentrations
• see which of the models fits best

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The steady-state approximation in enzyme kinetics

Two different mathematical formalisms for initial rate enzyme kinetics:

• 1. “rapid equilibrium” approximation

kchem << kd.S kchem << kd.P

• 2. “steady state” approximation

kchem ≈ kd.S kchem ≈ kd.P kchem > kd.S kchem > kd.P

• r

Analysis of initial rates in enzyme kinetics usually proceeds while invoking one of two theoretical formalisms, or approximations:

• “rapid equilibrium” approximation
• slow chemistry
• fast ligand dissociation
• 2. “steady state” approximation
• fast or slow chemistry
• it is a more general approach

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Importance of steady-state treatment: Therapeutic inhibitors

MANY ENZYMES THAT ARE TARGETS FOR DRUG DESIGN DISPLAY “FAST CHEMISTRY”

• T. Riera et al. (2008) Biochemistry 47, 8689–8696

Example: Inosine-5’-monophosphate dehydrogenase from Cryptosporidium parvum chemical step: fast hydride transfer

Arguably the “holy grail” of enzyme kinetics, in the context of therapeutic inhibition, is to understand the microscopic rate constants for enzyme-inhibitor interactions. However, such detailed understanding is not possible for “fast enzymes”, unless we actually invoke the steady-state approximation. The reason is that specifically the binding of inhibitors to enzyme-substrate complexes or enzyme-product complexes cannot be properly quantified without having a firm grasp of all microscopic rate constants in the given mechanism. (This is not true for “competitive” interactions, where the inhibitor binds to the free enzyme. For those interactions, it is enough to know the actual “on” and “off” rate constants.) Here is an example of an enzyme that has “fast” chemistry and so we must study the inhibition kinetics under the steady-state approximation.

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Steady-state initial rate equations: The conventional approach

The King-Altman method conventionally proceeds in two separate steps: Step One: Derive a rate equation in terms of microscopic rate constants Step Two: Rearrange the original equation in terms of secondary “kinetic constants”

• Measure “kinetic constants” (Km, Vmax, ...) experimentally.
• Compute micro-constant (kon, koff, ...) from “kinetic constants”, when possible.

EXPERIMENT:

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Steady-state initial rate equations: Example

• 2. derivation (“Step One”):

micro-constants

• 1. postulate a particular kinetic mechanism:

“kinetic” constants

• 3. rearrangement (“Step Two”):

Details: Segel, I. (1975) Enzyme Kinetics, Chapter 9, pp. 509-529.

• Whether or not the steady-state approximation is necessary

depends only on the relative magnitude of the chemical rate constants (red) vs. the dissociation rate constants (blue).

• Remember: The microscopic rates of the association steps (black)

depend on concentrations and therefore have no upper bound!

• After the first step of the derivation, the rate equation consists of

microscopic rate constants. Those are usually not accessible to direct measurement in initial-rate experiments.

• In the second step, the micro-constants are grouped in various ways

into “kinetic” constants such as Km nd kcat.

• Very soon we will see that this second step of the derivation is

not always possible. In contrast, the first step is always possible.

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Several problems with the conventional approach

1. Fundamental problem:

Step 2 (deriving “Km” etc.) is in principle impossible for branched mechanisms.

2. Technical problem:

Even when Step 2 is possible in principle, it is tedious and error prone.

3. Resource problem:

Measuring “kinetic constants” (Km, Ki, ...) consumes a lot of time and materials.

• 1. Fundamental problem:

Step 2 (rearrangement) is in principle impossible for branched mechanisms. The means: There can be no Km, Vmax, etc. derived for branched mechanisms. We can get a rate equation in terms of micro-constants, but then we are stuck.

• 2. Technical problem:

Even when Step 2 is possible in principle, it is tedious and error prone. These algebraic derivations typically are for “hard-core” kineticists only. It would be nice to automate all derivations and relegate this task to a machine.

• 3. Economics or resource management problem:

Measuring “kinetic constants” (Km, kcat, ...) consumes a lot of time and materials. Plus, it is not very clear how to convert “kinetic constants” to micro-constants. It would be nice to do a global fit, to extract as many micro-constants as possible.

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A solution to the fundamental problem

TURN THE CONVENTIONAL APPROACH ON ITS HEAD:

• Measure “kinetic constants” (Km, Vmax, ...) experimentally, when they do exist.
• Compute micro-constant (kon, koff, ...) from “kinetic constants”, when possible.

CONVENTIONAL APPROACH:

• Measure micro-constant (kon, koff, ...) experimentally.
• Compute “kinetic constants” (Km, Vmax, ...), when they do exist.

THE NEW APPROACH:

Advantages of the new approach: 1. Rate equations formulated in terms of micro-constants always exist. Rate equations in terms of “kinetic constants” do not always exist (branched mechanisms). 2. In some cases the best-fit values of micro-constants are actually their “true” values. Caveat: in other cases the best-fit micro-constants are only “apparent” values. But Even if we cannot estimate the “true” values (model redundancy, see below), we can

• ften at least estimate either the lower or the upper bound for a given micro-

constant. Disadvantages of the new approach: 1. In most cases the model formulated in terms of micro-constants is

• verparametrized (redundant).

However, model redundancy can be dealt with in numerous ways:

• make educated guesses based on literature reports;
• supply estimates from independent rapid-kinetic measurements;
• construct a minimal (“reduced”) kinetic model, with fewer steps

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A solution to the technical / logistical problem

USE A SUITABLE COMPUTER PROGRAM TO AUTOMATE ALL ALGEBRAIC DERIVATIONS

Kuzmic, P. (2009) Meth. Enzymol. 467, 247-280. INPUT: OUTPUT:

• The DynaFit software is available free of charge (to all academic

researchers and students) from the BioKin website.

• It has been cited in close to 900 journal articles up to this point,

most of them in the journal Biochemistry, followed by JBC.

• The “King-Altman” method in DynaFit has been beefed up in the last month or so,

specifically to facilitate the IMPDH collaboration with Liz Hedstrom’s group at Brandeis.

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A solution to the resource problem

USE GLOBAL FIT OF MULTI-DIMENSIONAL DATA TO REDUCE THE TOTAL NUMBER OF DATA POINTS

16-20 data points are sufficient

[mechanism] reaction S ---> P modifiers I E + S <==> E.S : ka.S kd.S E.S ---> E + P : kd.P E + I <==> E.I : ka.I kd.I ... [data] variable S ... file d01 | conc I = 0 | label [I] = 0 file d02 | conc I = 1 | label [I] = 1 file d03 | conc I = 2 | label [I] = 2 file d04 | conc I = 4 | label [I] = 4

DYNAFIT INPUT:

global fit

• Most of the time, we all do too many experiments (wastefully), because
• f the less-than-optimal methods for data analysis.
• The key is always use the global fit method.
• With global fit, we only need something like 3-5 times as many

data points as there are adjustable model parameters (rate constants).

• For example, if we are trying to determine 4 microscopic rate constants or

4 “kinetic constants”, then with a well-designed experiment we only need something like 16 or 20 data points.

• This is not true in the “traditional” approach to kinetic analysis:

analyzing subsets of data and then doing “re-plots” or “re-analyses”

• f intermediate results.

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Part II: Experiment

1. Background Inosine-5’-monophosphate dehydrogenase (IMPDH) and its importance 2. “Microscopic” kinetic model from stopped-flow data A complex “microscopic” model of IMPHD inhibition kinetics 3. Validating the transient kinetic model by initial rates Is our complex model sufficiently supported by initial-rate data?

Data: Dr. Yang Wei (Hedstrom Group, Brandeis University)

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IMPDH: Inosine-5’-monophosphate dehydrogenase

A POTENTIAL TARGET FOR THERAPEUTIC INHIBITOR DESIGN

Overall reaction: inosine-5’-monophosphate + NAD+ → xanthosine-5’-monophosphate + NADH

IMP “A” “B” XMP “P” “Q”

Chemical mechanism:

• This enzyme is important for drug design because the human form

is sufficiently different from the form found in pathogenic microorganisms.

• The human form is a “billion dollar” drug design target in its own right.

There are drugs currently in use targeting human IMPDH.

• A salient feature is that there is a covalent intermediate on the reaction

pathway.

• Importantly, many inhibitors seem to be binding almost exclusively to this

covalent intermediate. It turns out that there is no “rapid equilibrium” mechanism (and therefore a “rapid equilibrium” rate equation) that can properly account for this fact. Instead we really need steady-state.

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IMPDH kinetics: Fast hydrogen transfer catalytic step

HIGH REACTION RATE MAKES IS NECESSARY TO INVOKE THE STEADY-STATE APPROXIMATION very fast chemistry

• T. Riera et al. (2008) Biochemistry 47, 8689–8696

IMPDH from Cryptosporidium parvum

irreversible substrate binding

A = B = P = Q = IMP NAD+ XMP NADH UNITS: µM, sec

“rapid-equilibrium” initial rate equation should not be used

• Once again, here is the “fast chemistry” feature highlighted.
• Also note that the substrate “A” (i.e. NAD+) is binding effectively in an

irreversible fashion!

• That means that the dissociation rate constant is effectively zero.
• It can’t get any further from the classic “rapid equilibrium” approximation.
• In other words, the steady-state approximation is fundamentally

needed to analyze and understand the kinetics of these enzymes.

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Transient kinetic model for Bacillus anthracis IMPDH

THIS SCHEME FOLLOWS FROM STOPPED-FLOW (TRANSIENT) KINETIC EXPERIMENTS

A = B = P = Q = IMP NAD+ XMP NADH UNITS: µM, sec

very fast chemistry irreversible substrate binding

• Y. Wei, et al. (2015) unpublished

IMPDH from Bacillus anthracis

“rapid-equilibrium” initial rate equation should not be used

New (unpublished) stopped flow experiments produced this model. We’ll take it as a “given” for the purposes of this talk. DETAILS: Conditions:

• Very large excess of IMP (“A”) over Km(A): no free enzyme.
• Essentially a Uni-Uni kinetic mechanism (NAD+ → NADH, i.e. “B” → “Q”).
• The last step (E.P → E.S) is an irreversible product/substrate “exchange”.
• It is irreversible because it involves hydrolysis of a covalent intermediate.

Steady-state approximation is truly needed:

• The binding of NAD+ (“B”) to E.A is irreversible (Riera et al. [2008]).
• Hydride transfers is faster than any ligand dissociation, except NADH release.
• Binding of NAD+ (“B”) to E.P (substrate inhibition) is extremely slow.

Inhibition mode:

• Inhibitor (“I”) is essentially a product (NADH = “Q”) analog.
• It binds preferentially to the enzyme-XMP (“E.P”) complex, Kd ~90 nM.
• Somewhat weaker binding to the enzyme-IMP (“E.S”) complex, Kd ~1.4 µM.

Rapid equilibrium NADH release step (E.P.Q ↔ E.P):

• “On” constant set arbitrarily to 100 µM-1.s-1.

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Goal: Validate transient kinetic model by initial rate data

Two major goals:

1. Validate existing transient kinetic model Are stopped-flow results sufficiently supported by initial rate measurements? 2. Construct the “minimal” initial rate model How far we can go in model complexity based on initial rate data alone? Probing the IMPDH inhibition mechanism from two independent directions.

1. Validate existing transient kinetic model The transient kinetic model for B. anthracis IMPDH is highly complex: Is it sufficiently supported by initial rate measurements? 2. Construct the “minimal” initial rate model If we didn’t have the transient kinetic results, which “minimal” kinetic scheme would we end up with, based solely on initial rate measurements? Definition of a “minimal” model: Contains only the reaction steps (i.e. rate constants) that are actually defined by the available experimental data, no “extraneous” or “assumed” steps

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Three types of available initial rate data

1. Vary [NAD+] and [IMP]

substrate “B” and substrate “A”

2. Vary [NAD+] and [NADH] at saturating [IMP]

substrate “B” and product “Q” at constant substrate “A”

3. Vary [NAD+] and [Inhibitor] at saturating [IMP]

substrate “B” and inhibitor “I” at constant substrate “A”

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Simultaneous variation of [NAD+] and [IMP]

ADDED A NEW STEP – BINDING OF IMP (substrate “A”) TO THE ENZYME the only fitted rate constants

A = B = P = Q = IMP NAD+ XMP NADH UNITS: µM, sec

[IMP], µM

• All rate constants except ka.A / kd.A were fixed to values from

stopped-flow transient kinetics.

• It was necessary to add the E ↔ E.A step, because otherwise

we could not build a model for “A” (i.e., IMP) being varied in the experiment. CONCLUSIONS: The kinetic model predicted from the stopped-flow measurements agrees very well with the initial rate data.

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Simultaneous variation of [NAD+] and [NADH]

THIS CONFIRMS THAT NADH IS REBINDING TO THE E.P COMPLEX (“PRODUCT INHIBITION”)

A = B = P = Q = IMP NAD+ XMP NADH

Kd(NADH) = 90 µM

UNITS: µM, sec

[NADH], µM

• All rate constants were fixed to values from stopped-flow transient kinetics.

CONCLUSIONS: The kinetic model predicted from the stopped-flow measurements agrees very well with the initial rate data. NOTE: On the qualitative level, the stopped-flow model predicts that NADH (“Q”) does rebind to the enzyme-XMP complex (“E.P”). This prediction is clearly verified in the initial rate experiment. In general, product inhibition studies often are essential to a proper and complete understanding of many kinetic mechanisms.

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Simultaneous variation of [NAD+] and inhibitor [A110]

[Inh], µM

0.13

• All rate constants were fixed to values from stopped-flow transient kinetics

except the dissociation rate constant for “uncompetitive” inhibition, E.P.I → E.P + I.

• That rate constant had to be treated as an adjustable model parameter, in order

to achieve a sufficient goodness of fit.

• However, note that the difference between the postulated and fitted values

(0.27 s-1 vs. 0.13 s-1) is less than 50%.

• Such relatively small differences are not surprising, because the stopped-flow

experiment is conducted under experimental conditions (micromolar concentrations

• f enzyme) that are very different from the experimental conditions of the

initial rate measurements (nanomolar concentrations). CONCLUSIONS: The kinetic model predicted from the stopped-flow measurements agrees reasonably well with the initial rate data.

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Toward the “minimal” kinetic model from initial rate data

WHAT IF WE DID NOT HAVE THE STOPPED-FLOW (TRANSIENT) KINETIC RESULTS?

vary B + A vary B + Q vary B + I combine all three data sets, analyze as a single unit (“global fit”)

A global fit of enzyme kinetic data consists of combining all available data points into a single data set, regardless of which particular component (substrate, product, inhibitor) was varied in the experiment. This method of kinetic data analysis has proved superior (ref. [1]) to analyzing various subsets of the global data set, and subsequently “stitching together” various re-replots

• r re-fits of intermediate results.

In this case the individual subsets of data “report” on various distinct segments

• f the mechanism. For example the “B + Q” (i.e., NAD+ + NADH) data points

“report” exclusively on the rebinding of NADH to the enzyme-XMP complex. REFERENCES:

• Beechem, J. M.

“Global analysis of biochemical and biophysical data”

• Meth. Enzymol. 210 (1992) 37-54.

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The “minimal” kinetic model from initial rate data

INITIAL RATE AND STOPPED-FLOW MODELS ARE IN REASONABLY GOOD AGREEMENT

from initial rates from stopped-flow

0.27 0.30 Kd ≈ 70 Kd ≈ 90 Kd ≈ 6100 Kd ≈ 5800 Kd ≈ 0.04 Kd ≈ 0.09 15 15

UNITS: µM, sec

• The enzyme forms shown in red are those that can be “seen”

in both types of experiments: stopped-flow and initial rates.

• We cannot “see” the free enzyme (‘E’) in the stopped flow

experiment, because it is done at saturating concentration of IMP (‘A’).

• We cannot “see” the weakly bound complex E.A.I (Kd ~ 1 µM) in

the initial rate experiment, because the highest inhibitor concentration was very much lower than that ([I]max ~ 150 nM). To see this interaction, we’d have to increase [I]max >> 1 µM, but that is not practically achievable because by that time the enzyme is completely inhibited due to the other (“uncompetitive”) step.

• We cannot “see” the complex E.P.Q in the initial rate experiment,

because it is an isomeric form of another “central complex”, E.A.B. Distinct “central complexes” can never be observed at steady state.

• In the steady-state initial rate experiment we cannot see “on”

constants for any inhibition step (including substrate and product inhibition), so we can only compare equilibrium constants for those steps.

• With those caveats, the comparison turns out not too bad.

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The “minimal” kinetic model: derived kinetic constants

DYNAFIT DOES COMPUTE “Km” AND “Ki” FROM BEST-FIT VALUES OF MICRO-CONSTANTS

input

microscopic rate constants

primary output secondary output

derived kinetic constants

• “Best from both worlds”: DynaFit always reports the best-fit values of

microscopic rate constants. If “kinetic constants” can exist in principle (i.e., non-branched mechanism) we get those reported, too.

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The “minimal” kinetic model: derivation of kinetic constants

DYNAFIT DOES “KNOW” HOW TO PERFORM KING-ALTMAN ALGEBRAIC DERIVATIONS As displayed in the program’s output: automatically derived kinetic constants:

• The automatically derived King-Altman model is available

for inspection in the output files generated by DynaFit.

• This slide shows a screen shot of the relevant “model page”.

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Checking automatic derivations for B. anthracis IMPDH

“TRUST, BUT VERIFY”

• “Km” for NAD+:

turnover number, “kcat”:

• similarly for other “kinetic constants”
• Turnover number, kcat: It is an amalgamation of at least two

microscopic steps:

• 1. the NADH release step (kd.EP.Q) and
• 2. XMP release step (kd.P)
• However “XMP release” itself is an amalgamation of at least two

microscopic steps, not counting any conformational changes:

• 1. hydrolysis of the covalent intermediate and
• 2. XMP release proper.
• Also note that kcat = 12 s-1 is slower than the “rate limiting” step

(kd.P = 15 s-1). This is because “kinetic constants” are almost always – except for the simplest possible mechanisms – complex amalgamations of multiple rate constants.

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Reminder: A “Km” is most definitely not a “Kd”

+ B

A Kd is a dissociation equilibrium constant. However, NAD+ does not appear to dissociate.

kon = 2.7 × 104 M-1s-1 koff ≈ 0 Kd

(NAD)= koff/kon ≈ 0

Km

(NAD) = 450 µM

A Km sometimes is the half-maximum rate substrate concentration (although not in this case).

• The Michaelis constants (Km) is equal to the enzyme-substrate

complex dissociation constant (Kd) only if the chemical step is very slow compared to all ligand dissociations.

• In other words the Km = Kd equivalence only holds under

the “rapid equilibrium” approximation.

• However, in this case there is effectively no dissociation,

so there can be no “dissociation constant”.

• The best we can do (based on stopped-flow data) is to estimate

the upper limit of the enzyme-substrate dissociation rate constant.

• It is “very low” in this case, but it is hard to say just how low, for

various technical reasons. Other than that, the stopped-flow data are perfectly consistent with the dissociation constant being zero.

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Advantage of “Km”s: Reasonably portable across models

full model minimal model

kcat, s-1 Km(B), µM Ki(B), mM Ki(Q), µM Ki(I,EP), nM 13 430 6.6 77 50 12 440 7.4 81 45

turnover number Michaelis constant of NAD+ substrate inhibition constant of NAD+ product inhibition constant of NADH “uncompetitive” Ki for A110

• Although the “kinetic constants” (Km, kcat, Ki, ...) are not always

possible to obtain, when this is possible, they are quite useful.

• One advantage is that even if we get the microscopic model wrong,
• f have multiple equally plausible kinetic mechanisms, the

“kinetic constants” stay more-or-less the same.

• In that sense the kinetic constants have high “portability” across models.

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Reminder: A “Ki” is not necessarily a “Kd”, either ...

minimal model (initial rates): Kd = 37 nM King-Altman rate equation automatically derived by DynaFit: Kd

• We all learned at some point that the Michaelis constant is not

strictly speaking a dissociation equilibrium constant.

• However, it may not be equally widely recognized that, under the

steady-state approximation, the (external ligand) inhibition constants are not necessarily equilibrium dissociation constants, either.

• Some Ki’s are equal to the corresponding equilibrium constants,

but some are not. It just depends on the given mechanism.

• Here we see an example of a particular inhibition constant that

is not a simple ratio of the dissociation and association rate constants.

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... although in most mechanisms some “Ki”s are “Kd”s

full model (transient kinetics)

Kd = 1.3 µM King-Altman rate equation automatically derived by DynaFit: Kd

• The “competitive” Ki is a simple Kd
• The “uncompetitive” Ki is a composite

CONCLUSIONS:

• However, in the same kinetic mechanism as on the previous slide

we can find another inhibition constant that is, in fact, equivalent to a simple Kd for the given ligand.

• One of the rules of thumb (applicable especially to single-substrate

enzymes) is as follows:

• If the inhibitor binds to the free enzyme form, then the Ki for this

step is always identical to the corresponding Kd.

• But if the inhibitor binds to an enzyme-substrate or enzyme-product

complex, then the Ki for that step is not equal to the corresponding Kd.

• For multi-substrate enzymes the rules are not as clear,

as we can see here: binding to E.A vs. binding to E.P produces different results in terms of relationship between Ki and Kd.

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Part III: Summary and Conclusions

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Importance of steady-state approximation

• “Fast” enzymes require the use of steady-state formalism.
• The usual rapid-equilibrium approximation cannot be used.
• The same applies to mechanisms involving slow release of products.
• The meaning of some (but not all) inhibition constants depends on this.

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A “microscopic” approach to steady-state kinetics

• Many enzyme mechanisms (e.g. Random Bi Bi) cannot have “Km” derived for them.
• However a rate equation formulated in terms of micro-constants always exists.
• Thus, we can always fit initial rate data to the micro-constant rate equation.
• If a “Km”, “Vmax” etc. do actually exist, they can be recomputed after the fact.
• This is a reversal of the usual approach to the analysis of initial rate data.
• This approach combined with global fit can produce savings in time and materials.

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Computer automation of all algebraic derivations

• The DynaFit software package performs derivations by the King-Altman method.
• The newest version (4.06.027 or later) derives “kinetic constants” (Km, etc.) if possible.
• DynaFit is available from www.biokin.com, free of charge to all academic researchers.

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IMPDH kinetic mechanism

• IMPDH from B. anthracis follows a mechanism that includes NADH rebinding.
• This “product inhibition” can only be revealed if excess NADH is present in the assay.
• The inhibitor “A110” binds almost exclusively to the covalent intermediate.
• The observed inhibition pattern is “uncompetitive” or “mixed-type”

depending on the exact conditions of the assay.

• Thus a proper interpretation of the observed inhibition constant

depends on microscopic details of the catalytic mechanism. Note:

• Crystal structures of inhibitor complexes are all ternary: E·IMP·Inhibitor
• Therefore X-ray data may not show the relevant interaction.

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Acknowledgments

• Yang Wei

post-doc, Hedstrom group @ Brandeis All experimental data on IMPDH from Bacillus anthracis

• Liz Hedstrom

Brandeis University Departments of Biology and Chemistry